Tohoku Mathematical Journal

Some homological invariants of the mapping class group of a three-dimensional handlebody

Susumu Hirose

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Abstract

We show that, if g ≥ 2, the virtual cohomological dimension of the mapping class group of a three-dimensional handlebody of genus $\g$ is equal to $4\g-5$ and its Euler number is equal to 0.

Article information

Source
Tohoku Math. J. (2), Volume 55, Number 4 (2003), 543-549.

Dates
First available in Project Euclid: 11 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1113247129

Digital Object Identifier
doi:10.2748/tmj/1113247129

Mathematical Reviews number (MathSciNet)
MR2017224

Zentralblatt MATH identifier
1059.57009

Subjects
Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 20F38: Other groups related to topology or analysis 57M50: Geometric structures on low-dimensional manifolds

Keywords
Virtual cohomological dimension Euler numver 3-dimensional handlebody mapping class group

Citation

Hirose, Susumu. Some homological invariants of the mapping class group of a three-dimensional handlebody. Tohoku Math. J. (2) 55 (2003), no. 4, 543--549. doi:10.2748/tmj/1113247129. https://projecteuclid.org/euclid.tmj/1113247129


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References

  • J. S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure and Appl. Math. 22 (1969), 213--238.
  • K. S. Brown, Cohomology of groups, Grad. Texts in Math. 87, Springer-Verlag, New York-Berlin, 1982.
  • M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), 91--119.
  • H. B. Griffiths, Automorphisms of a 3-dimensional handlebody, Abh. Math. Sem. Univ. Hamburg 26 (1964), 191--210.
  • J. L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), 157--176.
  • J. L. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986), 457--485.
  • N. V. Ivanov, Subgroups of Teichmüller modular groups, Transl. Math. Monogr. 115, American Mathematical Society, Providence, RI, 1992.
  • N. V. Ivanov, Mapping class groups, Handbook of geometric topology, 523--633, North-Holland, Amsterdam, 2002.
  • G. Mess, Unit tangent bundle subgroups of the mapping class groups, preprint (1990).
  • D. McCullough, Twist groups of compact 3-manifolds, Topology 24 (1985), 461--474.
  • D. McCullough, Virtually geometrically finite mapping class groups of 3-manifolds. J. Differential Geom. 33 (1991), 1--65.
  • D. McCullough, private communication.
  • J.-P. Serre, Cohomology des groupes discrets, Prospects in Mathematics (Proc. Sympos., Princeton Univ., Princeton, N. J., 1970), 77--169, Ann. Math. Studies 70, Princeton Univ. Press, Princeton, N. J., 1971.